nonlinear dynamics and extension to three dimensions with externally ...

IOP PUBLISHING and INTERNATIONAL ATOMIC ENERGY AGENCY

NUCLEAR FUSION

Nucl. Fusion 50 (2010) 034007 (8pp)

doi:10.1088/0029-5515/50/3/034007

Hamiltonian four-field model for magnetic reconnection: nonlinear dynamics and extension to three dimensions with externally applied fields E. Tassi1 , P.J. Morrison2 , D. Grasso3,4 and F. Pegoraro5 1 Centre de Physique Th´eorique - UMR 6207, CNRS Luminy, Case 907, F-13288, Marseille Cedex 09, France 2 Department of Physics and Institute for Fusion Studies, University of Texas, Austin, TX 78712, USA 3 CNR-INFM, Genova, Italy 4 Burning Plasma Research Group, Dip. di Energetica, Politecnico di Torino, CNISM, Italy 5 Dipartimento di Fisica E. Fermi and CNISM, Pisa, Italy

E-mail: [email protected]

Received 15 May 2009, accepted for publication 10 August 2009 Published 23 February 2010 Online at stacks.iop.org/NF/50/034007 Abstract The nonlinear dynamics of a two-dimensional (2D) model for collisionless magnetic reconnection is investigated both numerically and analytically. For very low values of the plasma β, parallel magnetic perturbations tend to be proportional to the vorticity perturbations, but as β increases, detachment of these quantities takes place. The subsequent difference between the structure of the vorticity and the parallel magnetic perturbations can be explained naturally in terms of the ‘normal’ field variables that emerge from the noncanonical Hamiltonian theory of the model. A three-dimensional extension of the reconnection model is also presented, its Hamiltonian structure is derived, and the corresponding conservation properties are compared with those of the 2D model. A general method for extending a large class of 2D fluid plasma models to three dimensions, while preserving the Hamiltonian structure, is then presented. Finally, it is shown how such models can also be extended, while preserving the Hamiltonian structure, to include externally applied fields, that can be used, for instance, for modelling resonant magnetic perturbations. PACS numbers: 52.65.Kj, 96.60.Iv, 05.45.−a

(Some figures in this article are in colour only in the electronic version)

[4–6] and a 3D extension has been analysed in [7]. Many of the obtained results have relied on knowledge of the noncanonical Hamiltonian structure (see, e.g., [8]) of the model equations [3, 9]. Indeed, the Hamiltonian formulation led to the discovery of infinite families of Casimir invariants, associated with Lagrangian invariants. These invariants have made it possible to interpret and explain the formation of small scale structures in the plasma vorticity and current density [5]. A 2D, four-field model extension of the SPK model was obtained by Fitzpatrick and Porcelli (FP) in [10]. Unlike the SPK model, this model allows for magnetic and velocity perturbations in the direction of a guide field and also allows for finite β regimes, where as usual β is the ratio of the plasma and magnetic pressures. The Hamiltonian formulation of this model, derived in [11], proved to be useful for obtaining

1. Introduction Magnetic reconnection is believed to play a key role in many phenomena occurring in fusion plasmas, such as sawtooth oscillations in tokamaks and relaxation in reversed field pinches (see, e.g., [1, 2]). In high temperature tokamak plasmas with low collision frequencies, electron inertia can provide an effective mechanism for breaking the frozenin condition of magnetohydrodynamics and thereby allow magnetic reconnection to take place. In order to understand the nonlinear dynamics of this kind of magnetic reconnection driven by electron inertia, a three-field fluid model for collisionless plasmas was obtained by Schep, Pegoraro and Kuvshinov (SPK) [3]. A two-dimensional (2D), two-field version of this model has been intensively investigated in 0029-5515/10/034007+08$30.00

1

© 2010 IAEA, Vienna

Printed in the UK

Nucl. Fusion 50 (2010) 034007

E. Tassi et al

condition. The parameter dβ is defined as dβ = di cβ , with di corresponding to the ion skin depth. Equations (1)–(4) can be derived starting from a two-fluid description of a plasma. More precisely, equation (1) originates from the electron momentum equation, equation (2) from the electron vorticity equation, whereas equations (3) and (4) can be obtained from the plasma vorticity and parallel momentum equations, respectively. The system (1)–(4) has a noncanonical Hamiltonian formulation [11, 18], the Hamiltonian functional being given by
1 H = d2 x (de2 J 2 + |∇ϕ|2 + v 2 + |∇ψ|2 + Z 2 ), (5) 2 D

stability criteria [12], providing an unambiguous definition of negative energy modes [11], and interpreting numerical simulations [13, 14]. This paper is a continuation of this line of research, it being devoted to the investigation of magnetic reconnection in collisionless plasmas, while taking advantage of the Hamiltonian formalism. The purpose of the paper is twofold: first, results from a numerical investigation of the nonlinear dynamics of the parallel magnetic perturbations, perturbations that are not taken into account in the SPK model but are present in the FP model, are described. Second, it is shown how to extend a large class of 2D fluid models (which includes the FP model) to 3D Hamiltonian models with the inclusion of externally applied fields. The motivation for the latter is to include, within the fluid Hamiltonian framework, for instance, the effect of resonant magnetic perturbations that can have an important impact on turbulent fluctuations and transport (see, e.g. [15–17]). The paper is structured as follows. In section 2 the FP model and its Hamiltonian structure are briefly recalled. Results of the numerical simulations are then presented and their interpretation in terms of Hamiltonian ‘normal fields’ is given. In section 3, the 3D extension of the FP model is presented and its Hamiltonian structure is discussed. Next, the conditions for extensions of 2D models to 3D, with externally applied fields, are presented and a specific example is given. Finally, we conclude in section 4.

with J = −∇ 2 ψ, corresponding to the parallel current density and D indicating the spatial domain of interest. The quantity (5) corresponds to the total energy of the system and it naturally possesses a kinetic energy part, corresponding to the first three terms in (5), and a magnetic energy part, given by the remaining two terms. The noncanonical Hamiltonian formulation of the FP model is completed by a Poisson bracket, whose rather lengthy expression, in terms of the variables ψ −de2 ∇ 2 ψ, ∇ 2 ϕ, Z and v, can be found in [11]. We recall that this bracket has four infinite families of the so-called Casimir invariants, which are defined as functionals C of the field variables, that satisfy the relation {F, C} = 0, where {, } is the Poisson bracket and F is any functional of the field variables. Because they commute with any Hamiltonian functional, Casimir invariants are constants of motion for the system. The families of Casimirs for the FP model are
C1 = d2 x H(D), (6)

2. 2D four-field model 2.1. Four-field review In this section we first consider the cold-ion four-field model for magnetic reconnection in collisionless plasmas presented in [10]. This model is formulated in Cartesian coordinates (x, y, z), it assumes the presence of a strong guide field of amplitude B (0) directed along the ignorable coordinate z, and a constant background electron pressure P0 . The model equations, in a dimensionless form, are ∂(ψ − de2 ∇ 2 ψ) + [ϕ, ψ − de2 ∇ 2 ψ] − dβ [ψ, Z] = 0, ∂t

(1)

∂Z + [ϕ, Z] − cβ [v, ψ] − dβ [∇ 2 ψ, ψ] = 0, ∂t

(2)

∂∇ 2 ϕ + [ϕ, ∇ 2 ϕ] + [∇ 2 ψ, ψ] = 0, ∂t

(3)


C2 =

d2 xζ F (D),

(7)

d2 x g˜ + (T+ ) ,

(8)

d2 x g˜ − (T− ) ,

(9)


C3 =
C4 =

where H, F , g˜ + and g˜ − are arbitrary functions and D = ψe + di v, ζ = ∇ 2ϕ +

∂v + [ϕ, v] − cβ [Z, ψ] = 0, (4) ∂t where the bracket [, ] is defined by [f, g] = zˆ · (∇f × ∇g). Equations (1)–(2) describe the evolution of the fields ψ and Z, which determine the magnetic field through the √expression B (x, y, t) = ∇ψ × zˆ +(B (0) +cβ Z)ˆz, where cβ = β/(1 + β), 2 with β = (5/3)P0 /B (0) . The plasma velocity field v is determined by a stream function ϕ and a parallel component v by the expression v (x, y, t) = −∇ϕ × zˆ + v zˆ . The time evolution of the velocity follows from equations (3)–(4). The parameter de , appearing in (1), is the electron skin depth and is associated with the term responsible for breaking the frozen-in

T+ =

di Z, cβ d 2

di2 di di de v− Z, ψe − 2cβ d 3 2cβ d 2 2cβ d 3 de

(10) (11)

(12)

di2 di de di ψe + v− Z, (13) 3 3 2cβ d de 2cβ d 2cβ d 2  where ψe = ψ − de2 ∇ 2 ψ and d = di2 + de2 . The form of the Casimirs (6)–(9) leads to an alternative set of field variables, that we call ‘normal’ fields, according to the terminology introduced in [19], consisting of the variables D, ζ , T+ , T− , which are linear combinations of the original T− = −

2

Nucl. Fusion 50 (2010) 034007

E. Tassi et al

variables ψe , ∇ 2 ϕ, Z, v. In terms of the normal fields, the Poisson bracket of the FP model takes the simple form
{F, G} = d2 x (ζ [Fζ , Gζ ] + D([FD , Gζ ] + [Fζ , GD ]) + T− [FT− , GT− ] + T+ [FT+ , GT+ ]), where subscripts indicate functional derivatives. Hamiltonian (5) in terms of the normal fields becomes 
cβ2 d 4 2 D2 2 d x H = (T+ + T−2 ) + 2 2 2d di D

to its interpretation in terms of normal fields. In the following, we investigate the field Z and its dependence on the value of cβ , in order to see how the nonlinear evolution of Z develops in regimes with non-negligible values of cβ . Such a regime is not accessible in the SPK two-field model. Given that the FP model assumes p = −cβ Z, where p is an electron pressure perturbation, the evolution of Z can also provide information about the pressure evolution. Equations (1)–(4) were solved numerically using a code based on a finite volume scheme. The equations are solved on a rectangular domain {(x, y): −2π
x
2π, −π
y
π }, subdivided into 1024 × 1024 grid points, with periodic boundary conditions imposed for both the x and y directions. We chose an initial equilibrium of the form ψ0 (x) = 1/ cosh2 (x), ϕ0 = 0, Z0 = 0, v0 = 0, which is unstable to reconnecting perturbations. We are mainly interested in seeing how the evolution is modified, in comparison with the SPK model, when one increases cβ to nonnegligible values. Two cases are considered: the first case has a very low value of β, with the parameter dβ playing essentially the role of ρs of the SPK model, while the second case has higher β. In both cases the value of the normalized electron skin depth is set equal to 0.24. Note that, due to limitations in the numerical resolution, the electron/ion mass ratios for the two cases are slightly different. In figure 1 contour plots of ∇ 2 ϕ and Z, for cβ = 0.01, are compared. From the plots, it is evident that for these values the two fields are nearly identical. This is a direct consequence of considering a very low β with finite dβ ≈ ρs . As anticipated above, this suppresses the term cβ [v, ψ] in equation (2), which couples to the parallel dynamics, and allows for the solution Z = −dβ ∇ 2 ϕ. This requires the initial conditions for Z and ∇ 2 ϕ to be identical, which is the case for the simulations under consideration. It is interesting to note that the detachment of Z from −dβ ∇ 2 ϕ is measured by the amplitude of the normal field ζ . Indeed, for practical purposes, d ≈ di , and therefore ζ ≈ ∇ 2 ϕ+Z/dβ . In this regime, in which di is large, the source term in equation (17) becomes negligible and consequently, because ζ (x, y, 0) = 0, the value of ζ remains equal to zero at all times. In other words, the difference between Z and −dβ ∇ 2 ϕ cannot grow. Note also that, if ζ = 0, then

(14) The

1 − (ζ + T+ + T− ) ∇ −2 (ζ + T+ + T− ) 2   1 de D + c d(T − T ) − β + − 2 d2   de D + c d(T − T ) , (15) ×L β + − d2 where L is an operator defined by the relation Lψe = ψ. The normal fields are convenient variables that make the conservation laws of the system more evident. Indeed, by making use of the normal fields, the FP model can be written as ∂D = −[ϕ, D], ∂t ∂ζ = −[ϕ, ζ ] + d −2 [D, ψ], ∂t   ∂T± = − ϕ± , T± , ∂t where we have defined ϕ± := ϕ ±

cβ d ψ. de

(16) (17) (18)

(19)

From the compact form of equations (16)–(18), it possible to see that D and T± are Lagrangian invariants advected by velocity fields determined by the stream functions ϕ and ϕ± , respectively. The evolution equation for ζ , on the other hand, possesses a source term that vanishes when D is a flux function or when d → ∞. 2.2. Simulation results and interpretation

Z=−

The noncanonical Hamiltonian formulation and the knowledge of the Casimir invariants proved to be useful for investigating the nonlinear dynamics of the SPK reconnection model. From the knowledge of the Casimirs it was possible to deduce the conservation of the topology of field lines of the two Lagrangian invariants G± [4] and to explain the formation of small scale structures in the plasma vorticity [5]. The twofield version of the SPK model investigated in [4, 5] can be obtained from the FP model by taking the limit cβ → 0 and dβ → ρs , where ρs is the Larmor radius of ions with electron temperature. In this limit, equation (4) decouples from the system, whereas Z = −ρs ∇ 2 ϕ can be taken as the solution for Z, provided it is compatible with the initial conditions. From the expression Bz = (B (0) + cβ Z), one sees that the limit cβ → 0, corresponds to suppressing parallel magnetic perturbations. The evolution of such perturbations for the FP model has not yet been investigated, in particular, with regard

cβ d 2 (T+ + T− ), di

∇ 2 ϕ = T+ + T− .

(20)

Such a representation in terms of the Lagrangian invariants T± makes it possible to explain the formation of small scales in the vorticity and parallel magnetic field. As already explained in [12, 14] (and earlier in [5] for the two-field model) small scales are formed because of the stretching and filamentation of T+ and T− , advected by velocity fields that rotate in opposite directions. Figure 2, on the other hand, shows contour plots of ∇ 2 ϕ and Z for a higher value of β. In this case it is possible to see that the parallel magnetic field (or, equivalently, the pressure) and the vorticity follow distinct evolutions. Indeed, Z still undergoes a filamentation process, again corresponding to the stretching of the normal fields T+ and T− , whose structures are now even less regular than in the very low β regime. The vorticity, on the other hand, on top of the filamented structures, forms two vertical vortex sheets that collide and 3

Nucl. Fusion 50 (2010) 034007

E. Tassi et al

Figure 1. Contour plots of ∇ 2 ϕ (left column) and Z (right column). The top row refers to t = 50 Alfv`en times and the bottom row to t = 70 Alfv`en times. Values of the parameters are cβ = 0.01, dβ = 0.24 and de = 0.24. From the plots one can see that, because Z0 = ∇ 2 ϕ0 = 0, in this very low β regime, parallel magnetic perturbations closely track the vorticity perturbations.

Figure 2. Contour plots of ∇ 2 ϕ (left column) and Z (right column). The top row refers to t = 40 Alfv`en times and the bottom row to t = 55 Alfv`en times. Values of the parameters are cβ = 0.3, dβ = 0.72 and de = 0.24. In this higher β regime, the effect of the terms proportional to cβ is no longer negligible and Z detaches from −dβ ∇ 2 ϕ.

create vortices propagating in opposite directions along the y = 0 line. The superposition of these two types of dynamics is naturally explained in terms of the normal fields. Indeed, the relations Z=−

cβ d 2 (T+ + T− ), di

∇ 2 ϕ = ζ + T+ + T−

make it possible to see that it is the growth of the field ζ that prevents Z from being proportional to the vorticity in this β regime. The field ζ , in particular, is the one responsible for the formation of vortex sheets, that can eventually become unstable [12, 14, 20]. The pressure and the parallel magnetic field, on the other hand, do not receive contributions from ζ and therefore they do not form such localized structures. Instead,

(21) 4

Nucl. Fusion 50 (2010) 034007

E. Tassi et al

they continue to transfer energy to smaller and smaller scales, in a manner similar to what happens in a phase-mixing process.

compact form
3 {F, G} = d x ζ [Fζ , Gζ ] + D([FD , Gζ ] + [Fζ , GD ])

3. 3D four-field model

∂ ∂ GD + F D Gζ ∂z ∂z

di2 di2 ∂ ∂ + FT FT GT − GT . (26) 2cβ d 3 de + ∂z + 2cβ d 3 de − ∂z − +T− [FT− , GT− ] + T+ [FT+ , GT+ ] + Fζ

3.1. Extension of the four-field model We now assume that the field variables ψ, ϕ, Z and v also depend on the spatial coordinate z. We introduce a smallness parameter  defined as the ratio between the amplitudes of the poloidal magnetic field and of the strong guide magnetic field. We then assume the ordering   1  1, B (0) = O  ψ ∼ϕ∼Z∼v∼

The Hamiltonian, on the other hand, is still given by the expression of equation (15), with the obvious extension of the integral and the fields to three dimensions, but with ∇ replaced by ∇⊥ . In terms of the normal field variables the model equations become ∂D ∂ϕ = −[ϕ, D] − , ∂t ∂z ∂ζ = −[ϕ, ζ ] + d −2 [D, ψ] ∂t    de 1 ∂ , D − de L 2 D + cβ d(T+ − T− ) + d ∂z d

∂ ∂ ∼ = O(1), ∂x ∂y

∂ = O()  1, ∂z which is consistent with the ordering of the FP model. If one then carries out, with these assumptions, the derivation of the FP model as described in [10], the resulting 3D model equations are

  di2 ∂ ∂T± = − ϕ± , T± ± ∂t 2cβ d 3 de ∂z 

 2cβ2 d 4 de T± − ϕ ∓ cβ d L 2 D + cβ d(T+ − T− ) , × d di2

∂(ψ − de2 ∇⊥2 ψ) + [ϕ, ψ − de2 ∇⊥2 ψ] − dβ [ψ, Z] ∂t +

∂ϕ ∂Z = 0, + dβ ∂z ∂z

from which it transpires that, in 3D, the variables D, ζ , T+ and T− are no longer Lagrangian invariants. Casimirs for the 3D bracket (26) are functionals C such that   
∂CD {F, C} = d3 x Fζ [Cζ , ζ ] + [CD , D] + ∂z     di2 ∂CT+ ∂Cζ + FT+ [CT+ , T+ ] + + FD [Cζ , D] + ∂z 2cβ d 3 de ∂z   2 di ∂CT− =0 + FT− [CT− , T− ] − 3 2cβ d de ∂z

(22)

∂Z ∂v + [ϕ, Z] − cβ [v, ψ] − dβ [∇⊥2 ψ, ψ] − cβ ∂t ∂z −dβ

∂∇⊥2 ψ = 0, ∂z

(23)

∂∇⊥2 ϕ ∂∇⊥2 ψ = 0, + [ϕ, ∇⊥2 ϕ] + [∇⊥2 ψ, ψ] + ∂z ∂t

(24)

∂v ∂Z = 0, + [ϕ, v] − cβ [Z, ψ] − cβ ∂z ∂t

(25)

for all functionals F . This implies Casimirs are solutions of the following system:

where the perpendicular gradient operator is defined by ∇⊥ f = (∂f/∂x)xˆ + (∂f/∂y)y. ˆ Note that, with the above assumptions, the 3D extension of the model is obtained from the 2D version, by mapping the expressions [f, ψ], for a generic field f , into [f, ψ] + ∂f/∂z. The model (22)–(25) can describe magnetic reconnection in three dimensions, in which case the Hamiltonian that describes the finite-dimensional system for magnetic field lines, ψ(x, y, z, t), is in general non-integrable and can therefore lead to chaotic magnetic field lines (see, e.g. [21]). The cβ → 0, dβ → ρs limit of (22)–(25) is also consistent with the 3D version of the two-field SPK model, which was investigated numerically in [7, 14, 22]. Our 3D model is consistent with the four-field system recently derived in [23]. Analogously to the 2D version, this 3D model also admits a noncanonical Hamiltonian formulation. The latter can be conveniently expressed by making use of the normal fields of equations (10)–(13) after replacing ∇ with ∇⊥ . Indeed, in these variables, the bracket for the 3D model has the relatively

[Cζ , ζ ] + [CD , D] +

∂CD = 0, ∂z

(27)

∂Cζ = 0, ∂z

(28)

[CT+ , T+ ] +

di2 ∂CT+ = 0, 2cβ d 3 de ∂z

(29)

[CT− , T− ] −

di2 ∂CT− = 0. 2cβ d 3 de ∂z

(30)

[Cζ , D] +

Equation (28), for instance, is of finite-dimensional Hamiltonian form, with D playing the role of the Hamiltonian, z playing the role of time, and the unknown Cζ playing the role of the dynamical variable. Because we seek a solution for the case where D depends on z, this is a ‘time-dependent’ Hamiltonian system. For general D there is no solution of this equation besides the initial conditions, and these are not useful because they are not isolating, i.e. they do not define surfaces. 5

Nucl. Fusion 50 (2010) 034007

E. Tassi et al

The extension to three dimensions, therefore, eliminates the infinite families of Casimirs that were present in the 2D version. Casimir solutions can, however, still be found and are given by


C2 = d3 x ζ, C3 = d3 x T+ , C1 = d3 x D,
(31) C4 = d3 x T− .

and this imposes restrictions on the elements of the matrix Aij . In the following we derive these restrictions. ij First, we obviously assume that the Wk have the commutation properties (see [29]) necessary for the Jacobi identity of {F, G}⊥ , i.e. we assume {{F, G}⊥ , H }⊥ + cyc ≡ 0,

where cyc indicates the addition of the two terms obtained by cyclic permutation of F , G and H . By a theorem in [28], it follows for the form of {F, G} given by (34) that

The 3D model admits an additional constant of motion that is not a Casimir given by the functional  
d3 2 3 2 (32) h = d x ζ D + cβ de 2 (T+ − T− ) , di which, in terms of the original fields, takes the form    
Z h = d3 x ∇⊥2 ϕ ψe + di ∇⊥2 ϕ + v . cβ

{{F, G} , H } + cyc ≡ 0. {{F, G}, H } + cyc = {{F, G}⊥ , H }⊥ + {{F, G}⊥ , H } + {{F, G} , H }⊥ + {{F, G} , H } + cyc.

(33)

(38)

The first and last terms of (38) vanish by virtue of (36) and (37). The third term vanishes because δ{F, G} = ˙ 0, δχ i

(39)

where = ˙ means modulo second variation terms. Thus, it remains to show what the condition

3.2. General extensions The 3D extension carried out for the FP model in section 3.1 follows a path first travelled in [24], where an analogous extension was carried out for reduced magnetohydrodynamics (RMHD) [25]. The procedure for going from 2D to 3D can be formalized and applied to a large class of 2D models. Of particular interest for plasma physics is a class of 2D magnetofluid models, where the presence of a strong guide field is assumed. This class of models includes, besides the SPK, FP and RMHD, the four-field model for tokamak dynamics of Hazeltine et al [26], the model for ion-temperature-gradient driven and drift Kelvin– Helmholtz modes of Waelbroeck et al [27] or the more recent electromagnetic gyrofluid model of Waelbroeck et al [19]. Indeed, for all such models, the extension to 3D amounts to extending a 2D Lie–Poisson bracket by including the dependence of the fields on the coordinate that was ignorable in the original 2D version. The general form of the extended Poisson bracket, for the class that includes all these models, has the form

{{F, G}, H } + cyc = {{F, G}⊥ , H } + cyc ≡ 0

(40)

implies. Using δ{F, G}⊥ ij = ˙ Wk [Fi , Gj ], δχ k we obtain

(41)




˙ {{F, G}⊥ , H } + cyc =

d3 x Ars Wrij ([Fi , Gj ]∂Hs

+ [Gi , Hj ]∂Fs + [Hi , Fj ]∂Gs ).

(42)

Integrating the first term of (42) by parts gives
{{F, G}⊥ , H } + cyc = ˙ d3 x Ars Wrij (−[∂Fi , Gj ]Hs −[Fi , ∂Gj ]Hs + [Gi , Hj ]∂Fs + [Hi , Fj ]∂Gs ).

(43)

The first and third terms of (43) give
d3 x Ars Wrij (−[∂Fi , Gj ]Hs + [Gi , Hj ]∂Fs )
=

d3 x [Gi , Hj ]∂Fs (Ars Wrij − Arj Wrsi ), (44) where the ‘f gh-identity’, f [g, h] = − g[f, h], has been used and indices have been shifted. Similarly, the remaining two terms of (43) give
d3 x Ars Wrij (−[Fi , ∂Gj ]Hs + [Hi , Fj ]∂Gs )

(34)

where ∂ := ∂/∂z, assuming that z was the ignorable coordinate, and where χ 1 , · · · χ n indicate the field variables ij ji for an n-field model. Note Aij = Aj i and Wk = Wk , which are required for the bracket to be antisymmetric. The bracket is thus the sum of the bracket of the original 2D model, {F, G}⊥ , with a bracket that accounts for the 3D extension, {F, G} . Not all antisymmetric bilinear operators of the form (34) are Poisson brackets, even if {F, G}⊥ is a Poisson bracket. In fact, a Poisson bracket must also satisfy the Jacobi identity (see, e.g. [8, 28]) {{F, G}, H } + {{H, F }, G} + {{G, H }, F } = 0,

(37)

Therefore

Note, if Z = −dβ ∇⊥2 ϕ, this quantity reduces to the generalized cross-helicity which is conserved in the 3D two-field SPK model [9].

{F, G} = {F, G} + {F, G}⊥

ij = d3 x Aij Fi ∂Gj + d3 x Wk χ k [Fi , Gj ] ,

(36)


=

d3 x [Hi , Fj ]∂Gs (Ars Wrij − Ari Wrj s ).

(45)

Because {{F, G}, H } + cyc ≡ 0 for all functionals F, G, H , we obtain from (44) and (45) the following compatibility conditions on Aij , needed for the Jacobi identity: Ars Wrij = Arj Wrsi = Ari Wrj s .

(35) 6

(46)

Nucl. Fusion 50 (2010) 034007

E. Tassi et al

In the specific case of the FP model, if we choose χ = D,

χ = ζ,

1

χ = T+ ,

2

3

(47)

where ψext is the external magnetic perturbation. This leads to the following model: ∂(ψ − de2 ∇⊥2 ψ) + [ϕ, ψ − de2 ∇⊥2 ψ] − dβ [ψ, Z] ∂t ∂ϕ ∂Z + + dβ = −dβ de2 [ψext , Z], ∂z ∂z

(48)

∂Z ∂v ∂∇ 2 ψ + [ϕ, Z] − cβ [v, ψ] − dβ [∇⊥2 ψ, ψ] − cβ − dβ ⊥ ∂t ∂z ∂z

χ = T− , 4

and then obtain  0 1  A= 0 0

1 0

0 0

0 0

a 0

 0 0    0  −a

= −dβ [ψext , ψ] + dβ de2 [ψext , ∇⊥2 ψ] + cβ de2 [ψext , v]

with a = di2 /(2cβ d 3 de ) and the only non-vanishing elements ij of Wk are W222 = W112 = W121 = W333 = W444 = 1.

−dβ

∂∇⊥2 ϕ ∂∇⊥2 ψ + [ϕ, ∇⊥2 ϕ] + [∇⊥2 ψ, ψ] + ∂t ∂z ∂ψext = [ψext , ψ] − de2 [ψext , ∇⊥2 ψ] + , ∂z ∂Z ∂v + [ϕ, v] − cβ [Z, ψ] − cβ ∂t ∂z

(49)

Therefore, condition (46) is satisfied and (26) is indeed a Poisson bracket. 3.3. Addition of external field perturbations

= cβ de2 [ψext , Z]. Evidently, there are two kinds of drive here, additive and multiplicative, and so there are various possibilities. For example, one can temporally drive a 2D theory additively by supposing (56) ψext = z (x, y, t),

The 3D four-field model (22)–(25) can be extended to include the presence of external perturbations in such a way that the resulting extended model can be shown to still admit a Hamiltonian formulation. Indeed, we can obtain a system driven by an external perturbation by adding to the Hamiltonian a term of the form
j (50) Hext = d3 x χ i Dij χext (x, y, z, t),

with some prescribed 2D perturbing magnetic flux. We point out that such extension to externally driven Hamiltonian models, obtained by using (50), can be applied to all Hamiltonian plasma models with Poisson brackets, and in particular those of the form of (34).

j

where χext indicates the external fields and (Dij ) is a constant matrix. Keeping the same Poisson bracket, the model equations for the externally perturbed system are then given by ∂χ i = {χ i , H + Hext }. ∂t

4. Conclusions

(51)

In this paper, different aspects of the nonlinear dynamics and modelling of magnetic reconnection in collisionless plasmas have been considered. In section 2 we addressed the question of seeing whether the vorticity phase-mixing process and the layer formations, which were observed in a simpler model valid at very low β, persist at higher β values, which can be accessed by the FP model. Numerical simulations of the FP model showed that at very low values of β, and when dβ ≈ ρs , also parallel magnetic perturbations undergo a phase-mixing process in which energy is transferred to small scales. On the other hand, we also showed that this process tends to disappear when higher β regimes are entered. This behaviour was naturally explained in terms of the growth of the normal field ζ , which also measures the deviation from proportionality between the parallel magnetic perturbation and the vorticity. In section 3, we presented an extension of the FP model to three dimensions. We proved that the 3D extension still possesses a Hamiltonian structure, but that the number of Casimir invariants was dramatically reduced. Next, a condition was derived for determining 3D extensions for a large class of 2D plasma models, which preserve the Hamiltonian structure. Finally, we showed how such 3D Hamiltonian models can also account for externally applied fields. Numerical simulations of such extended models in the presence of external fields, and the investigation of the resulting nonlinear dynamics, will be a subject of future work.

It is important to note that with this form for the drive, the Casimir invariants remain constants of motion, because it still follows that (52) {C, H + Hext } = 0, i.e. the drive allows one to reach only dynamically accessible states. Energy is added to (or subtracted from) the system according to the following: dH = {H, Hext }, dt and




Hext =

dt

dHext = dt




dt

(53)

 ∂Hext + {Hext , H } . (54) ∂t

As an example, we present an extended version of the 3D four-field model that takes into account the effect of external magnetic poloidal perturbations and that still possesses the Hamiltonian structure. This can be obtained by considering the Hamiltonian
1 d3 x (de2 J 2 + |∇ϕ|2 H¯ = H + Hext = 2 D + v 2 + |∇ψ|2 + Z 2 + ψext ψe ),

∂ψext , ∂z

(55) 7

Nucl. Fusion 50 (2010) 034007

E. Tassi et al

[14] Grasso D., Borgogno D., Pegoraro F. and Tassi E. 2009 Coupling between reconnection and Kelvin–Helmholtz instabilities in collisionless plasmas Nonlinear Process. Geophys. 16 241–9 [15] Reiser D. and Scott B. 2005 Electromagnetic fluid drift turbulence in static ergodic magnetic fields Phys. Plasmas 12 122308 [16] Reiser D. 2007 Impact of large island perturbations on turbulent blob transport in tokamaks Phys. Plasmas 14 082314 [17] Reiser D. and Chandra D. 2009 Plasma currents induced by resonant magnetic field perturbations in tokamaks Phys. Plasmas 16 042317 [18] Tassi E., Morrison P.J. and Grasso D. 2007 Hamiltonian structure of a collisionless reconnection model valid for high and low β plasmas Collective Phenomena in Macroscopic Systems ed G. Bertin et al (Singapore: World Scientific) pp 197–206 [19] Waelbroeck F.L., Hazeltine R.D. and Morrison P.J. 2009 A Hamiltonian electromagnetic gyrofluid model Phys. Plasmas 16 032109 [20] Del Sarto D., Califano F. and Pegoraro F. 2003 Secondary instabilities and vortex formation in collisionless-fluid magnetic reconnection Phys. Rev. Lett. 91 235001 [21] Morrison P.J. 2000 Magnetic field lines, Hamiltonian dynamics, and nontwist systems Phys. Plasmas 7 2279–89 [22] Grasso D., Borgogno D. and Pegoraro F. 2007 Secondary instabilities in two and three-dimensional magnetic reconnection in fusion relevant plasmas Phys. Plasmas 14 055703 [23] Blan N.H. and Tsiklauri D. 2009 Compressible Hall magnetohydrodynamics in a strong magnetic field Phys. Plasmas 16 064503 [24] Morrison P.J. and Hazeltine R.D. 1984 Hamiltonian formulation of reduced magnetohydrodynamics Phys. Fluids 27 886–97 [25] Strauss R. 1976 Nonlinear three-dimensional magnetohydrodynamics of noncircular tokamaks Phys. Fluids 19 134 [26] Hazeltine R.D., Hsu C.T. and Morrison P.J. 1987 Hamiltonian four-field model for nonlinear tokamak dynamics Phys. Fluids 30 3204–11 [27] Waelbroeck F.L., Morrison P.J. and Horton W. 2004 Hamiltonian formulation and coherent structures in electrostatic turbulence Plasma Phys. Control. Fusion 46 1331–50 [28] Morrison P.J. 1982 Poisson brackets for fluids and plasmas Mathematical Methods in Hydrodynamics and Integrability in Dynamical Systems (American Institute of Physics Conference Proceedings vol 88) ed M. Tabor and Y. Treve (New York: American Institute of Physics) pp 13–45 [29] Thiffeault J.L. and Morrison P.J. 2000 Classification and Casimir invariants of Lie–Poisson brackets Physica D 136 205–44

Acknowledgments This work was supported by the European Community under the contract of Association between EURATOM, CEA, and the French Research Federation for fusion studies. The views and opinions expressed herein do not necessarily reflect those of the European Commission. Financial support was also received from the Agence Nationale de la Recherche (ANR EGYPT) and PJM was supported by the US Department of Energy Contract No. DE-FG03-96ER-54346. We acknowledge useful discussions with the Nonlinear Dynamics group at the Centre de Physique Th´eorique.

References [1] Biskamp D. 2000 Magnetic Reconnection in Plasmas (Cambridge: Cambridge University Press) [2] Priest E.R. and Forbes T.G. 2000 Magnetic Reconnection (Cambridge: Cambridge University Press) [3] Schep T.J., Pegoraro F. and Kuvshinov B.N. 1994 Generalized two-fluid theory of nonlinear magnetic structures Phys. Plasmas 1 2843–51 [4] Cafaro E., Grasso D., Pegoraro F., Porcelli F. and Saluzzi A. 1998 Invariants and geometric structures in nonlinear Hamiltonian magnetic reconnection Phys. Rev. Lett. 80 4430–3 [5] Grasso D., Califano F., Pegoraro F. and Porcelli F. 2001 Phase mixing and saturation in Hamiltonian reconnection Phys. Rev. Lett. 86 5051–4 [6] Porcelli F., Borgogno D., Califano F., Grasso D., Ottaviani M. and Pegoraro F. 2002 Recent advances in collisionless magnetic reconnection Plasma Phys. Control. Fusion 44 389–406 [7] Borgogno D., Grasso D., Califano F., Farina D., Pegoraro F. and Porcelli F. 2005 Aspects of three-dimensional magnetic reconnection Phys. Plasmas 12 032309 [8] Morrison P.J. 1998 Hamiltonian description of the ideal fluid Rev. Mod. Phys. 70 467–521 [9] Kuvshinov B.N., Pegoraro F. and Schep T.J. 1994 Hamiltonian formulation of low-frequency, nonlinear plasma dynamics Phys. Lett. A 191 296 [10] Fitzpatrick R. and Porcelli F. 2004 Collisionless magnetic reconnection with arbitrary guide field Phys. Plasmas 11 4713–8, 14 049902 (erratum) [11] Tassi E., Morrison P.J., Waelbroeck F.L. and Grasso D. 2008 Hamiltonian formulation and analysis of a collisionless fluid reconnection model Plasma Phys. Control. Fusion 50 085014 [12] Tassi E., Grasso D., Pegoraro F. and Morrison P.J. 2008 Stability and nonlinear dynamics aspects of a model for collisionless magnetic reconnection J. Plasma Fusion Res. Ser. at press [13] Tassi E., Grasso D. and Pegoraro F. 2010 Nonlinear processes in Hamiltonian reconnection Commun. Nonlinear Sci. Numer. Simul. 15 2

8